﻿ 基于操纵运动方程的水动力导数计算方法研究
 舰船科学技术  2017, Vol. 39 Issue (1): 26-31 PDF

Research on calculation method of hydrodynamic derivatives based on maneuvering equation
WU Xing-ya, GAO Xiao-peng
Department of Naval Architecture Engineering, Naval University of Engineering, Wuhan 430033, China
Abstract: On STAR-CCM+ software platform, the article uses the RANS equations and VOF method, a numerical simulation on PMM movement for a specific model, considering the free liquid surface and the attitude of the model change in the process of oblique towing, maneuverability hydrodynamic derivative solving method is established with three conditions, including drifting motions, swaying motions and yawing motions of different oscillation modes, the results of simulation calculation was compared with the results by regression equation，confirmed the effectiveness of the method to solve the hydrodynamic derivatives based on the software platform of STAR-CCM+.
Key words: maneuvering equation     hydrodynamic derivatives     STAR-CCM+     curve fitting
0 引言

1 数值计算方法 1.1 控制方程

 $\frac{{\partial {{\bar u}_i}}}{{\partial {x_i}}} = 0 ,$ (1)
 $\begin{array}{l} {\rm{\rho }}\frac{{\partial {{\bar u}_i}}}{{\partial t}} + \rho {{\bar u}_j}\frac{{\partial {u_i}}}{{\partial {x_j}}} =- \frac{{\partial {{\bar p}_i}}}{{\partial {x_i}}} + \frac{\partial }{{\partial {x_j}}}\left( {\mu \frac{{\partial {{\bar u}_i}}}{{\partial {x_j}}}- \rho \overline {u_i'u_j'} } \right) \end{array}。$ (2)

1.2 湍流模型

 $\begin{array}{l} {\rm{\rho }}\frac{{dk}}{{dt}} = \frac{\partial }{{\partial {x_j}}}\left[ {\left( {\mu + \frac{{{\mu _t}}}{{{\sigma _k}}}} \right)\frac{{\partial k}}{{\partial {x_j}}}} \right] + {G_k} + {G_b}- \rho \varepsilon- {Y_M} \end{array},$ (3)
 $\begin{array}{l} {\rm{\rho }}\frac{{d\varepsilon }}{{dt}} = \frac{\partial }{{\partial {x_i}}}\left[ {\left( {\mu + \frac{{{\mu _t}}}{{{\sigma _\varepsilon }}}} \right)\frac{{\partial \varepsilon }}{{\partial {x_i}}}} \right] + \rho {C_1}S\varepsilon- \\[5pt] \, \, \quad \quad \rho {C_2}\frac{{{\varepsilon ^2}}}{{k + \sqrt {v\varepsilon } }} + {C_{1\varepsilon }}\frac{\varepsilon }{k}{C_{3\varepsilon }}{G_b}。 \end{array}$ (4)

1.3 VOF 算法及 STAR-CCM+ 的求解原理应用

VOF 是一个简单的多项流模型，通过 VOF 算法中 VOF 波的设定，在三维情况下，使船模适用于 6 自由度运动模型。VOF 算法是一种处理自由面的方法，其求解原理是根据各个时刻流体在网格单元体积的变动量与网格单元自身体积的比值函数F来构造、追踪自由面，确定自由面的形状和位置。当在某一时刻网格单元中比值函数F = 1 时，说明该时刻状态下该网格单元均被指定相的流体充满，当F = 0 时，说明该单元均被另一相流体充满。F 函数满足方程：

 $\frac{{\partial F}}{{\partial t}} + u\frac{{\partial F}}{{\partial x}} + v\frac{{\partial F}}{{\partial y}} + w\frac{{\partial F}}{{\partial z}} = 0。$ (5)

2 数值计算模型 2.1 计算对象

 图 1 计算船模模型 Fig. 1 Calculation model of ship

2.2 数值计算域

 图 2 船模计算区域 Fig. 2 Calculation area of model
2.3 数值计算条件设定

2.4 数值计算网格划分

 图 3 计算控制域网格划分 Fig. 3 Meshing in control domain
3 水动力导数计算模型及计算结果 3.1 斜行运动计算模型

 图 4 斜航运动 Fig. 4 Drifting motion
 $r = 0,$ (6)
 $\nu =- V\sin \beta 。$ (7)

 $\nu =- V\sin \beta \approx- V\beta 。$ (8)

 ${Y} = {Y_V}\nu + {Y_{VVV}}{\nu ^3},$ (9)
 ${N} = {N_V}\nu + {N_{VVV}}{\nu ^3},$ (10)

 $Y' = \frac{Y}{{\frac{1}{2}\rho {V^2}{L^2}}},$ (11)
 $N' = \frac{N}{{\frac{1}{2}\rho {V^2}{L^3}}},$ (12)
 $\nu ' = \frac{\nu }{V} =- \sin \beta。$ (13)

 图 5 不同侧向速度下侧向力曲线 Fig. 5 Curve of side force in different side velocity
 $y\!=\!-2.89 \times {10^{\!-\!6}}\!-\!0.01736x\!+\!0.00101{x^2}\!-\!0.23733{x^3},$ (14)

 图 6 不同侧向速度下首摇力矩曲线 Fig. 6 Curve of yaw moment in different side velocity
 $y\!=\!1.86\!\times\!{10^{\!-\!7}}\!-\!0.00589x\!+\!6.13205\!\times\!{10^{\!-\!5}}\!{x^2}\!+\!0.0104{x^3}。$ (15)

3.2 纯首摇运动计算模型

 图 7 纯首摇运动 Fig. 7 Yawing motion
 $\psi = {\psi _0}\cos \omega t,$ (16)
 ${\psi _0} =- \frac{{a\omega }}{V};$ (17)
 $r = \dot \psi =- {\psi _0}\omega \sin \omega t,$ (18)
 $\dot r = \ddot \psi =- {\psi _0}{\omega ^2}\cos \omega t。$ (19)

 ${Y} = {Y_r}r + {Y_{\dot r}}\dot r + {Y_{rrr}}{r^3},$ (20)
 ${N} = {N_r}r + {N_{\dot r}}\dot r + {N_{rrr}}{r^3}。$ (21)

 $\begin{array}{l} Y' = Y_r'\left( {\frac{{a{\omega ^2}L}}{{{V^2}}}} \right)\sin \omega t + Y_{\dot r}'\left( {\frac{{a{\omega ^3}{L^2}}}{{{V^3}}}} \right)\cos \omega t + \\[5pt] \quad \quad Y_{rrr}'\frac{1}{6}{\left( {\frac{{a{\omega ^2}L}}{{{V^2}}}} \right)^3}{\left( {\sin \omega t} \right)^3}{\rm{ = }}\\[5pt] \quad \quad A\sin \omega t + B\cos \omega t + C{\left( {\sin \omega t} \right)^3}, \end{array}$ (22)
 $\begin{array}{l} N' = N_r'\left( {\frac{{a{\omega ^2}L}}{{{V^2}}}} \right)\sin \omega t + N_{\dot r}'\left( {\frac{{a{\omega ^3}{L^2}}}{{{V^3}}}} \right)\cos \omega t + \\[5pt] N_{rrr}'\frac{1}{6}{\left( {\frac{{a{\omega ^2}L}}{{{V^2}}}} \right)^3}{\left( {\sin \omega t} \right)^3}{\rm{ = }} M\sin \omega t + N\cos \omega t + P{\left( {\sin \omega t} \right)^3}, \end{array}$ (23)

 ${{A}} = Y_r'\left( {\frac{{a{\omega ^2}L}}{{{V^2}}}} \right),$ (24)
 ${{B}} = Y_{\dot r}'\left( {\frac{{a{\omega ^3}{L^2}}}{{{V^3}}}} \right),$ (25)
 ${{C}} = Y_{rrr}'\frac{1}{6}{\left( {\frac{{a{\omega ^2}L}}{{{V^2}}}} \right)^3},$ (26)
 ${{M}} = N_r'\left( {\frac{{a{\omega ^2}L}}{{{V^2}}}} \right),$ (27)
 ${{N}} = N_{\dot r}'\left( {\frac{{a{\omega ^3}{L^2}}}{{{V^3}}}} \right),$ (28)
 ${{P}} = N_{rrr}'\frac{1}{6}{\left( {\frac{{a{\omega ^2}L}}{{{V^2}}}} \right)^3}。$ (29)

3.2.1 方案 1（固定幅值取不同首摇频率）下计算结果

 图 8 振幅a=0.15m,船舶首摇频率0.16Hz工况下的船舶重力曲线

 $\begin{split} \\[-12pt] y =& 0.0008424\sin \omega t + 0.003464\cos \omega t + \\ & 0.0001224{\left( {\sin \omega t} \right)^3}; \end{split}$ (30)
 $\begin{split} \\[-12pt] y =& 0.00065\sin \omega t + 0.003279\cos \omega t + \\& 0.000103{\left( {\sin \omega t} \right)^3}。 \end{split}$ (31)

 图 9 不同振荡频率下纯首摇水动力导数Y′r 拟合结果 Fig. 9 Hydrodynamic derivativesY′r fitting result of yawingmotion in different oscillatory frequency

 图 10 不同振荡频率下纯首摇水动力导数N′r 拟合结果 Fig. 10 Hydrodynamic derivativesN′r fitting result of yawing motion in different oscillatory frequency

3.2.2 方案 2 （固定频率取不同振荡幅值）下计算结果

 图 11 不同振幅下纯首摇水动力导数Y′r 拟合结果 Fig. 11 Hydrodynamic derivativesY′r fitting result of yawing motion in different amplitude

 图 12 不同振幅下纯首摇水动力导数N′r 拟合结果 Fig. 12 Hydrodynamic derivativesN′r fitting result of yawing motion in different amplitude

4 结语

1） 通过该方法数值计算所得的线性水动力导数与采用回归公式计算结果在较小误差范围内，两者吻合度较好，证实了基于 STAR-CCM+ 软件平台采用 PMM 运动数值模拟计算线性水动力导数的可靠性与准确性。

2）采用固定幅值下不同振荡频率与固定频率下不同幅值两种纯首摇运动工况求取水动力导数，两者计算结果相吻合，进一步证实了本文方法计算船舶特定水动力导数的真实性。

3）基于本文方法对船舶水动力导数的可靠求解，为下一步的船舶操纵性仿真预报的开展奠定了基础。

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